AdequacyModel (version 1.0.8)

goodness.fit: Adequacy of models

Description

This function provides some useful statistics to assess the quality of fit of probabilistic models, including the statistics Cramer-von Mises and Anderson-Darling. These statistics are often used to compare models not fitted. You can also calculate other goodness of fit such as AIC, CAIC, BIC, HQIC and Kolmogorov-Smirnov test.

Usage

goodness.fit(pdf, cdf, starts, data, method="L-BFGS-B", domain=c(0,Inf),
             mle=NULL)

Arguments

pdf

Probability density function;

cdf

Cumulative distribution function;

starts

Initial parameters to maximize the likelihood function;

data

Data vector;

method

Method used for minimization of the function -log(likelihood). The methods supported are: L-BFGS-B (default), BFGS, Nelder-Mead, SANN, CG. Can also be transmitted only the first

domain

Domain of probability density function. By default the domain of probability density function is the open interval 0 to infinity.This option must be an vector with two values;

mle

Vector with the estimation maximum likelihood. This option should be used if you already have knowledge of the maximum likelihood estimates. The default is NULL, ie, the function will try to obtain the estimates of maximum likelihoods.

Value

• W
  Statistic Cramer-von Misses;
• A
  Statistic Anderson Darling;
• KS
  Kolmogorov Smirnov test;
• mle
  Maximum likelihood estimates;
• AIC
  Akaike Information Criterion;
• CAIC
  Consistent Akaikes Information Criterion;
• BIC
  Bayesian Information Criterion;
• HQIC
  Hannan-Quinn information criterion;
• Erro
  Standard errors of the maximum likelihood estimates;
• Value
  Minimum value of the function -log(likelihood);
• Convergence
  0 indicates successful completion and 1 indicates that the iteration limit maxit had been reached. More details at optim.

encoding

latin1

Details

The function goodness.fit returns statistics KS (Kolmogorov-Smirnov), A (Anderson-Darling), W (Cramer-von Misses). Are also calculated other measures of goodness of fit. These functions are: AIC (Akaike Information Criterion), CAIC (Consistent Akaikes Information Criterion), BIC (Bayesian Information Criterion) and HQIC (Hannan-Quinn information criterion).

The Kolmogorov-Smirnov test may return NA with a certain frequency. The return NA informs that the statistical KS is not reliable for the data set used. More details about this issue can be obtained from ks.test.

By default, the function calculates the maximum likelihood estimates. The errors of the estimates are also calculated. In cases that the function can not obtain the maximum likelihood estimates, the change of the values initial, in some cases, resolve the problem. You can also enter with the maximum likelihood estimation if there is already prior knowledge.

References

Chen, G., Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, 154-161.

Nocedal, J. and Wright, S. J. (1999) Numerical Optimization. Springer.

Sakamoto, Y., Ishiguro, M., and Kitagawa G. (1986). Akaike Information Criterion Statistics. D. Reidel Publishing Company.

Hannan, E. J. and Quinn, B. G. (1979). The Determination of the Order of an Autoregression. Journal of the Royal Statistical Society, Series B, 41, 190-195.

See Also

For details about the optimization methodologies may view the functions optim, ks.test, nlminb.

Examples

# Example 1:

data(carbone)

# Exponentiated Weibull - Probability density function.
pdf_expweibull <- function(par,x){
  beta = par[1]
  c = par[2]
  a = par[3]
  a * beta * c * exp(-(beta*x)^c) * (beta*x)^(c-1) * (1 - exp(-(beta*x)^c))^(a-1)
}

# Exponentiated Weibull - Cumulative distribution function.
cdf_expweibull <- function(par,x){
  beta = par[1]
  c = par[2]
  a = par[3]
  (1 - exp(-(beta*x)^c))^a
}

goodness.fit(pdf=pdf_expweibull, cdf=cdf_expweibull, 
             starts = c(1,1,1), data = carbone,
             method="L-BFGS-B", domain=c(0,Inf),mle=NULL)
             
# Example 2:

data = c(0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63,
              0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40,
              2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50,
              2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51,
              2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81,
              2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64,
              3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69,
              4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69,
              4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75,
              4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 
              5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62,
              7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93,
              11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13,
              1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51,
              6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07,
              21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69)

# Kumaraswamy Weibull Poisson - Probability density function.
pdf_kwweibullpoisson <- function(par,x){
  a = par[1]
  b = par[2]
  c = par[3]
  lambda = par[4]
  beta = par[5]
  (a*b*c*lambda*(beta^c)*(x^(c-1))*((1-exp(-(x*beta)^c))^(a-1)) *
   ((1-(1-exp(-(beta*x)^c))^a)^(b-1)) * 
   exp(-lambda*(1-(1-(1-exp(-(beta*x)^c))^a)^b)
   -  (beta*x)^c))/(1-exp(-lambda))
}

# Kumaraswamy Weibull Poisson - Cumulative distribution function.
cdf_kwweibullpoisson <- function(par,x){
  a = par[1]
  b = par[2]
  c = par[3]
  lambda = par[4]
  beta = par[5]
  (1 - exp(lambda*(-(1-(1-(1-exp(-(x*beta)^c))^a)^b))))/(1-exp(-lambda)) 
}

goodness.fit(pdf=pdf_kwweibullpoisson, cdf=cdf_kwweibullpoisson, 
             starts = c(0.120,1.010,1.000,1.000,0.100), data = data,
             method="L-BFGS-B", domain=c(0,Inf),mle=NULL)

data = rweibull(250,2,3)
goodness.fit(pdf=pdf_kwweibullpoisson, cdf=cdf_kwweibullpoisson, 
             starts = c(0.120,1.010,1.000,1.000,0.100), data = data,
             method="L-BFGS-B", domain=c(0,Inf),mle=NULL)

# Example 3:

# Kumaraswamy Beta - Probability density function.
pdf_kwbeta <- function(par,x){
  beta = par[1]
  a = par[2]
  alpha = par[3]
  b = par[4]
  (a*b*x^(alpha-1)*(1-x)^(beta-1)*(pbeta(x,alpha,beta))^(a-1)*
  (1-pbeta(x,alpha,beta)^a)^(b-1))/beta(alpha,beta) 
}
# Kumaraswamy Beta - Cumulative distribution function.
cdf_kwbeta <- function(par,x){
  beta = par[1]
  a = par[2]
  alpha = par[3]
  b = par[4]
  1 - (1 - pbeta(x,alpha,beta)^a)^b
}

data = rbeta(1000,2,2.2)

goodness.fit(pdf = pdf_kwbeta, cdf=cdf_kwbeta, starts=c(0.5,1.02,2,1),
data=data, method="L-BFGS-B", domain=c(0,1), mle=NULL)